Conference article

Implementation of the Contensou-Erismann Model of Friction in Frame of the Hertz Contact Problem on Modelica

Ivan Kosenko
Russian State University of Tourism and Service, Department of Engineering Mechanics, Russia

Evgeniy Aleksandrov
Russian State University of Tourism and Service, Department of Engineering Mechanics, Russia

Download article

Published in: Proceedings of the 7th International Modelica Conference; Como; Italy; 20-22 September 2009

Linköping Electronic Conference Proceedings 43:31, s. 288-298

Show more +

Published: 2009-12-29

ISBN: 978-91-7393-513-5

ISSN: 1650-3686 (print), 1650-3740 (online)


An approximate model to compute resulting wrench of the dry friction tangent forces in frame of the Hertz contact problem is built up. An approach under consideration develops in a natural way the contact model constructed earlier. Generally an analytic computation of the integrals in the Contensou–Erismann model leads to the cumbersome calculation; decades of terms; including rational functions depending in turn on complete elliptic integrals. To implement the elastic bodies contact interaction computer model fast enough one builds up an approximate model in the way initially proposed by Contensou.

To verify the model built results obtained by several authors were applied. First the Tippe-Top dynamic model is used as an example under testing. It turned out the top revolution process is identical to one simulated with use of the set-valued functions approach. In addition; the ball bearing dynamic model was also used to verify different approaches to the tangent forces computational implementation in details. A model objects corresponding to contacts between balls and raceways were replaced by ones of a new class developed here. Then the friction model of the approximate Contensou type embedded into the whole bearing dynamic model was thoroughly tested.


Hertz contact model; Contensou simplified model; Contensou–Erismann model; Vil’ke model; Tippe-Top; ball bearing model


[1] Contensou; P.; Couplage entre frottement de glissement et frottement de pivotement dans la théorie de la toupie. In: Kreiselprobleme Gyrodynamics: IUTAM Symposium Celerina; 1962; Berlin: Springer; 1963; pp. 201–216.

[2] Erismann; Th.; Theorie und Anwendungen des echten Kugelgetriebes. Z. angew. Math. Phys.; 1954; Vol. 5; No. 5; pp. 355–388. doi: 10.1007/BF01600548.

[3] Kosenko I. I.; Alexandrov E. B.; Implementation of the Hertz Contact Model and Its Volumetric Modification on Modelica. In: Bachmann; B. (Ed.)Proceedings of the 6th International Modelica Conference; Bielefeld; Germany; March 3–4; 2008; Bielefeld: The Modelica Association; and University of Applied Sciences Bielefeld; 2008; pp. 203–212.

[4] Kossenko; I. I.; Implementation of Unilateral Multibody Dynamics on Modelica. In: Schmitz; G. (Ed.) Proceedings of the 4th International Modelica Conference; Hamburg–Harburg; Germany; March 7–8; 2005; Hamburg–Harburg: The Modelica Association; and The Department of Thermodynamics; Hamburg University of Technology; 2005; pp. 13–23.

[5] Hertz; H.; Über die Berührung fester elastischer Körper. J. reine und angewandte Mathematik; 1882; B. 92; S. 156–171.

[6] Landau; L. D. and Lifshitz; E. M.; Theory of Elasticity. 3rd Edition. Landau and Lifshitz Course of Theoretical Physics. Volume 7. Oxford – Boston – Johannesburg – Melbourne – New Delhi – Singapore: Reed Educational and Professional Publishing Ltd.; 1999.

[7] Leine; R. I. and Nijmeijer; H.; Dynamics and Bifurcations of Non-Smooth Mechanical Systems. Berlin – Heidelberg – New York: Springer Ver-lag; 2004. doi: 10.1007/978-3-540-44398-8.

[8] Novozhilov; I. V.; Conditions of Stagnation in Systems with the Coulomb Friction. Mechanics of Solids; 1973; Vol. 8; No. 1; pp. 8–14.

[9] Novozhilov; I. V.; Fractional Analysis: Methods of Motion Decomposition; Boston: Birkhauser; 1997.

[10] Zhuravlev; V. F.; The Model of Dry Friction in the Problem of the Rolling of Rigid Bodies. J. Appl. Math. Mech.; 1998; Vol. 62; No. 5; pp. 705–710. doi: 10.1016/S0021-8928(98)00090-2.

[11] Rooney; G. T. and Deravi; P.; Coulomb Friction in Mechanism Sliding Joints. Mechanism and Machine Theory; 1982; Vol. 17; Iss. 3; pp. 207–211. doi: 10.1016/0094-114X(82)90006-4.

[12] Kireenkov; A. A.; Three-Dimensional Model of Combined Dry Friction and Its Application in Non-Holonomic Mechanics. In: van Campen; D. H.; Lazurko; M. D. van den Oever; W. P. J. M.; (Eds.) Proceedings of ENOC2005; Fifth EUROMECH Nonlinear Dynamics Conference; Eindhoven; August 7–12; 2005; Eindhoven; The Netherlands: Eindhoven University of Technology; 2005.

[13] Aleksandrov; E. B.; Vil’ke; V. G. Kosenko; I. I.; Hertzian Contact Problem: Numerical Reduction and Volumetric Modification. Computational Mathematics and Mathematical Physics; 2008; Vol. 48; No. 12; pp. 2226–2240. doi: 10.1134/S0965542508120129.

[14] Whittaker; E. T.; Watson; G. N.; A Course of Modern Analysis; Cambridge–New York–Mel-bourne – Madrid – Cape Town: Cambridge University Press; 2002.

[15] Janke; E.; Emde; F.; Lösch; F.; Tafeln Höherer Funktionen; Stuttgart: B. G. Teubner Verlagsgesellschaft; 1960.

[16] Milne-Thomson; L. M.; Elliptic Integrals; Abramowitz; M.; Stegun; I. A.; (Eds) Handbook of Mathematical Functions: with Formulas; Graphs; and Mathematical Tables; New York: Dover Publications Inc.; 1972.

[17] Bateman; H.; Erdélyi; A.; Higher Transcendental Functions. Volume 3. New-York – Toronto – London: Mc Graw-Hill Book Company; Inc.; 1955.

[18] Kosenko; I. I.; Loginova; M. S.; Obraztsov; Ya. P. and Stavrovskaya; M. S.; Multibody Systems Dynamics: Modelica Implementation and Bond Graph Representation. In: Kral; Ch. and Haumer; A.; (Eds.) Proceedings of the 5th International Modelica Conference; Vienna; Austria; September 4–5; 2006; Vienna: The Modelica Association; and arsenal research; 2006; pp. 213–223.

[19] Leine; R. I. and Glocker; Ch.; A Set-Valued Force Law for Spatial Coulomb–Contensou Friction. Europian Journal of Mechanics A/Solids; 2003; Vol. 22; No. 2; pp. 193–216.

[20] Friedl; C.; Der Stehaufkreisel. Master’s thesis. Augsburg: Institut für Physik; Universität Augsburg; 1997.



Citations in Crossref